In many cases,
\(\pi \)  is only
approximately Markov with respect to a
sparse graph \(\mathcal {G}\) , e.g., there might exist
a second
density
\(\pi ^{\prime }\)  which is Markov with respect to \(\mathcal {G}\)  and
such that \(\mathcal {D}_{\rm KL}(\pi ^{\prime } \vert \vert \pi ) < \varepsilon \) , for some \(\varepsilon >0\) .
In particular,
we know that the filtering
marginals
will not be “exactly” Markov with respect to a sparse
\(\mathcal {G}\) ,
since conditional independence properties do not persist over time for general models [1]}.
Yet, when filtering spatially distributed processes (e.g., in data assimilation) it is reasonable to expect that the filtering (or forecast) marginals will be
well-approximated by sparse MRFs, since the latter
encode probabilistic models with local or
neighboring interactions.
For example, in Section  we consider the Lorenz-96 model, which arises from the discretization of a one-dimensional PDE on a periodic domain.
In that case, we can expect that the filtering marginal is approximately Markov with respect to a cycle
graph with a low-degree of connectivity between the nodes. When dealing with two-dimensional PDEs, it is
natural to consider grid graphs [2]}, [3]}, or more general lattices on the plane, and so on for higher dimensional domains.
In practice, we can
then
sparsify the estimator
\(\widehat{S}\)  according to a graph \(\mathcal {G}\) , which implies Markov properties for \(\pi \)  that are only
approximately satisfied, and treat
the exact geometry of the graph—e.g., its neighborhood structure—as a calibration parameter of the algorithm, as if it were a localization radius.
This is equivalent to
regularizing the estimation of the KR rearrangement by
projecting \(\pi \)  onto a manifold of sparse MRFs
[1]}.
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